Limit Cycles for Non Smooth Differential Equations via Schwarzian Derivative

We consider a class of planar polynomial systems with discontinuous righthand sides and prove that, under certain hypotheses, it presents at most one singular limit cycle and two regular limit cycles. Furthermore the sum of the multiplicity of the regular limit cycles is less or equal than two. A key point in the proof is the study of the Schwarzian derivative of the return map. Finally, we give some examples reaching these bounds.